Potential Energy, Energy Conservation, and Conservative Forces: Study Notes

Potential Energy & Energy Conservation

Introduction

Potential energy is the energy associated with the position of bodies in a system. This section covers gravitational and elastic forces as examples of conservative forces, the properties of potential energy functions, and the derivation of force from potential energy.

Gravitational Potential Energy

Gravitational potential energy (PE) is defined for a body of mass m at height y above a reference point:

• Definition: $U_{grav} = mgy$

• Work Done by Gravity: When a body moves downward, gravity does positive work and PE decreases. When a body moves upward, gravity does negative work and PE increases.

Equations:

• Downward motion: $W_{grav} = mg(y_1 - y_2) > 0$, $\Delta U_{grav} = mgy_2 - mgy_1 = -W_{grav} < 0$

• Upward motion: $W_{grav} = -mg(y_2 - y_1) < 0$, $\Delta U_{grav} = mgy_2 - mgy_1 = -W_{grav} > 0$

Conclusion: $W_{grav} = -\Delta U_{grav}$

Note: Gravitational PE acts like a bank to store work for later use. It is only meaningful for the system of the body and the Earth.

Work-Energy Theorem & Conservation of Mechanical Energy

The work-energy theorem relates the change in kinetic energy to the work done by all forces:

• $\Delta K = -\Delta U_{grav}$ (if only gravity acts)

• $K_{initial} + U_{grav, initial} = K_{final} + U_{grav, final}$

• Conservation of Mechanical Energy: In the absence of non-conservative forces, the total mechanical energy (kinetic + potential) is conserved.

Elastic Potential Energy (Spring)

Elastic potential energy is stored in a spring when it is stretched or compressed:

• Work Done by Spring: $W_{el} = \int F_{spring} dx = \frac{1}{2}kx_1^2 - \frac{1}{2}kx_2^2$

• Elastic PE of Spring: $U_{el} = \frac{1}{2}kx^2$

• For gravitational PE, zero level position is arbitrary; for elastic PE, zero level must correspond to unstretched position.

Combined Gravitational and Elastic Forces

When both gravitational and elastic forces act, the work-energy theorem becomes:

• $W_{grav} + W_{el} + W_{other} = \Delta K$

• $K_{initial} + (U_{grav, initial} + U_{el, initial}) = K_{final} + (U_{grav, final} + U_{el, final})$

• If $W_{other} = 0$, mechanical energy is conserved.

Application Example: Elevator and Spring

For an elevator with a broken cable and friction, the spring constant k can be found using energy conservation:

• $W_{other} = -fd$ (work done by friction)

• $\Delta K = 0 - \frac{1}{2}mv_i^2$

• $\Delta PE = -mgd + \frac{1}{2}kd^2$

• $k = \frac{2(mgd + \frac{1}{2}mv_i^2 - fd)}{d^2}$

Conservative and Non-Conservative Forces

Conservative Forces

Work done by conservative forces can be "reclaimed" as kinetic or potential energy. Examples include gravitational and spring forces.

• Properties:

  1. Work can be expressed as the difference between initial and final values of a potential energy function.

  2. It is reversible; if the path is reversed, work done changes sign.

  3. Depends only on starting and ending points, not the path.

  4. Work done around a closed loop is zero.

• Non-Conservative Forces: Work done by friction cannot be "reclaimed"; it is path-dependent and dissipative.

Testing for Conservative Forces

• Check if the work done around a closed loop is zero.

• Work done by friction is path-dependent and always negative.

Deriving Force from Potential Energy

• In 1D: $F = -\frac{dU}{dx}$

• In 3D: $F_x = -\frac{\partial U}{\partial x}$, $F_y = -\frac{\partial U}{\partial y}$, $F_z = -\frac{\partial U}{\partial z}$

• Potential energy functions can be shifted by a constant without changing the force.

Energy Diagrams and Equilibrium

Stable and Unstable Equilibrium

Energy diagrams help interpret the stability of equilibrium points:

• Stable Equilibrium: Potential energy is at a minimum; small displacements result in restoring forces.

• Unstable Equilibrium: Potential energy is at a maximum; small displacements result in forces that move the particle away from equilibrium.

Example: A hypothetical potential-energy function $U(x)$ shows minima (stable) and maxima (unstable) points.

Practice Questions & Applications

Conceptual and Quantitative Questions

• Questions cover gravitational and elastic potential energy, work-energy theorem, energy conservation, and interpretation of energy diagrams.

• Example: Determining the speed of a ball thrown upward using energy conservation.

• Example: Analyzing the motion of a block on a frictionless incline and spring.

• Example: Interpreting potential energy graphs to identify points of maximum speed, acceleration, and equilibrium.

Summary Table: Conservative vs Non-Conservative Forces

Type of Force	Work Path Dependence	Energy Recovery	Examples
Conservative	No (depends only on endpoints)	Yes	Gravity, Spring
Non-Conservative	Yes (depends on path)	No	Friction, Air Resistance

Key Equations

• Gravitational PE: $U_{grav} = mgy$

• Elastic PE: $U_{el} = \frac{1}{2}kx^2$

• Work-Energy Theorem: $\Delta K = W_{total}$

• Conservation of Mechanical Energy: $K_{initial} + U_{initial} = K_{final} + U_{final}$

• Force from Potential Energy: $F = -\frac{dU}{dx}$

Additional info: Some diagrams and questions refer to textbook figures and inverted text for answers, but the main concepts and equations are fully covered above for self-contained study.